Electric field Maxwell equations

Taking the electric field from equation (3-246) and substituting into another Maxwell equation. 

Under the conditions of a static electric field, the Maxwell s field equations reduce to the Poisson equation [111,112,172,261], given by... 

To illustrate the use of the vector operators described in the previous section, consider the equations of Maxwell. In a vacuum they provide the basic description of an electromagnetic field in terms of the vector quantifies the electric field and 9C the magnetic field The definition of the field in a dielectric medium requires the introduction of two additional quantities, the electric displacement SH and the magnetic induction. The macroscopic electromagnetic properties of the medium are then determined by Maxwell s equations, viz. 

The electromagnetic field in free space is described by the electric field vector E and the magnetic field vector H, which in the absence of charges satisfy Maxwell s equations... 

Solving Maxwell s equations using Equations (6.1) and (6.2) in the slowly varying envelope approximation, one can calculate the amplitude of the electric field of the TH to be (Ward and New 1969 Bjorklund 1975) ... 

In order to understand the diffuse layer in detail, we need to go back to the fnndamental eqnations of electrostatics due to J.C. Maxwell. The equation of interest relates the local electric field E(r) at the position vector r to the net local electric charge density p(r) ... 

Consider a plane wave propagating in a nonabsorbing medium with refractive index N2 = n2, which is incident on a medium with refractive index A, = w, + iky (Fig. 2.4). The amplitude of the incident electric field is E(, and we assume that there are transmitted and reflected waves with amplitudes E, and Er, respectively. Therefore, plane-wave solutions to the Maxwell equations at... 

The Maxwell-Heaviside theory of electrodynamics has no explanation for the Sagnac effect [4] because its phase is invariant under 7 as argued already, and because the equations are invariant to rotation in the vacuum. The d Alembert wave equation of U(l) electrodynamics is also 7 -invariant. One of the most telling pieces of evidence against the validity of the U(l) electrodynamics was given experimentally by Pegram [54] who discovered a little known [4] cross-relation between magnetic and electric fields in the vacuum that is denied by Lorentz transformation. 

Ion bombardment rate is determined from ion momentum or continuity equations, depending upon assumptions made in the model. To solve equations for ion and electron momentum and energy balances, the electric field profile must be known. This profile is obtained from the governing Maxwell equation, which is usually Poisson s equation. 

In the present simple example, Eq. (32) immediately suggests a valid solution namely, that the magnetic field must lie along the y axis, thus leading to the well-known orthogonality between the electric and magnetic fields. A bona fide solution for Maxwell s equations is then provided by the electric field of Eq. (30), and... 

From the symmetric set, an extended set of Maxwell equations was exhibited in Section V.E. This set contains currents and sources for both fields E, B. The old conjecture of Dirac s is vindicated, but the origin of charge density is always electric (i.e., no magnetic monopole). Standard Maxwell s equations are a limiting case in far field. 

TE) and transverse magnetic (TM) parts. However, Rumsey [53] detailed a secondary method of solving the same equations that effected a decomposition of the field into left-handed and right-handed circularly polarized parts. For such unique field solutions to the time-harmonic Maxwell equations (e = electric permittivity, p = magnetic permeability) ... 

This is clearly a Beltrami equation, but what is more amazing is that the field result (88) describes a solution to the free-space Maxwell equations that, in contrast to standard PWS, the electric (E0) and magnetic (Bo) vectors are parallel [e.g., Eo x Bo = 0, where Eo x Bo = i(E0 A Bo)], the signal (group) velocity of the wave is subluminal (v < c), the field invariants are non-null, and as (91) clearly shows, this wave is not transverse but possesses longitudinal components. Moreover, Rodrigues and Vaz found similar solutions to the free-space Maxwell equations that describe a superluminal (v > c) situation [71]. 

It should be mentioned that in the approach with nonzero electric divergence, the photon mass is also related to the space charges in vacuo. Now, in the approach with a / 0, we have j = ctE but jeff = 0. Let us now assume j = aE and j 7 0, which means fs 0. In such a case, jo is assumed to be associated with p, where p is the charge density in vacuo. So, in such an approach one can think of the existence of a kind of space charge in vacuo that is to be considered to be associated to nonzero electric field divergence. This will result in a displacement current in vacuum similar to that measured by Bartlett and Corle [43]. The assumption of the existence of space charge in vacuo makes our theory not only fully relativistic but also helps us to understand gauge condition. In the conventional framework of Maxwell s equations... 

With the connection of PDEs, and especially soliton forms, to group symmetries established, one can conclude that if the Maxwell equation of motion that includes electric and magnetic conductivity is in soliton (SGE) form, the group symmetry of the Maxwell field is SU(2). Furthermore, because solitons define Hamiltonian flows, their energy conservation is due to their symplectic structure. 

The Fourier transform S (w, b) is found by solving the Maxwell equations. In the case of a fast particle with charge ze moving with constant speed, the components ( , b) and < ft(w, b) of its electric field ( Jv and <9fc b) are given by the formulas148... 

We start from the first pair of Maxwell equations written with account of electric charges moving in vacuum. Let vector J be the electric current density produced by these charges. Combining the above-mentioned equations, we get the second-order differential equation for electric field vector E ... 

We start from Maxwell equations. Using the MKS system of units, we have for wave vector k perpendicular to electric field E the equation [see Eq. (3)]... 

Electromagnetic waves combine the propagation of two vector fields, E and B. These are the electric and magnetic induction fields, respectively, and in a vacuum are governed by the Maxwell equations 1,2,3] ... 

Since the Maxwell s equations are linear in E and H, the solution to this problem is easily solved by decomposing the fields into their x and y components. The electric field is... 

It is instructive to develop the solution for scattering by a small sphere of radius, a X. In such a limit the sphere is represented as a point dipole, and to determine its polarizability, the interaction of the sphere with the electric field is modeled as shown in Figure 4.5. The restriction that the sphere is much smaller that the wavelength of light suggests that to a first approximation, the electric field, at an instant in time, appears to the sphere as a uniform field. We must solve the following time-independent Maxwell s equations [1],... 

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